To be completely explicit, we explain the labels on the
three classes of maps in the theorem.
A map \(g\colon X\rightarrow Y\)  in \({K}/B\)  is called an \(f\) -equivalence
if it is a fiberwise homotopy equivalence. This needs the notion of
homotopy over \(B\) , which is formulated in terms of
\(X\times _B I\) .
A map \(g\colon X\rightarrow Y\)  in \({K}/B\)  is called an \(f\) -fibration
if it has the fiberwise covering homotopy property, i.e. if it has the
RLP property with respect to all maps of the form
\(i_0\colon Z\rightarrow Z\times _B I\)  for all \(Z\in {K}/B\) . A map \(g\colon X\rightarrow Y\) 
in \({K}/B\)  is called an \(\bar{f}\) -cofibration, or a strong cofibration
if it has the LLP property with respect to all \(f\) -acyclic \(f\) -fibrations.
There is also the notion of an \(f\) -cofibration: this is a map
\(g\colon X\rightarrow Y\)  which satisfies the LLP property with respect to all maps of the form
\(p_0\colon \mathrm {Map}_B(I,Z)\rightarrow Z\)  for some \(Z\in {K}/B\) .
Every \(\bar{f}\) -cofibration \(g\colon X\rightarrow Y\)  in \({K}/B\)  is an
\(f\) -cofibration. The converse is not true in general. However May
and Sigurdsson prove (see Theorems 4.4.4 and 5.2.8 of [1]})
that if \(g\colon X\rightarrow Y\)  is a closed \(f\) -cofibration then \(g\)  is an \(\bar{f}\) -cofibration.
